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LVIII. Methods in the Integral Calculus. By the Rev, Rosert 
CarmicHArEL, A.M., Fellow of Trinity College, Dublin*. 
ke N interesting exemplification of the reciprocal aid which 
geometry and analysis lend to each other is afforded by 
the fact, that there is a large class of differential equations whose 
integration may be much facilitated by the employment of geo- 
metrical considerations, more especially those associated with the 
transformation to polar coordinates. It is obvious, too, that 
where differential equations admit of geometrical interpretations, 
such interpretations are more likely to be suggested in this way 
than where their integration is conducted by methods purely 
analytical. 
Thus, as a simple example, if it be proposed to integrate the 
equation 
zdx + ydy=m(ydx —xdy), 
by transformation to polar coordinates we get at once ~ 
rdr=mrd0, or dr=mrde ; 
whence, instantly, the curve denoted by the given equation is 
the logarithmic spiral 
r=Ce™, 
or, in rectangular coordinates, 
(a2 +y)'=Cem™ 
The immediate geometrical interpretation of the given equa- 
tion is easily seen by dividing both sides of its transformed type 
by ds. In fact, then, we get 
i. dr =n F 
diz. alae s 
and, if Q be the intercept between the point on the curve and the 
foot of the perpendicular from the origin on the tangent (P), this 
equation is at once equivalent to 
Q=mP ; 
or the given differential equation represents, as is known, the 
plane curve for which the perpendicular from the origin on the 
tangent is in a constant proportion to the intercept between the 
corresponding point of contact and the foot of the perpendicular. 
The curve which satisfies this condition is, as we haye seen, the 
logarithmic spiral. wigs, 
Thus the immediate geometrical interpretation of the given 
equation, as well as the more remote, are with equal facility ob- 
* Communicated by the Author. 
